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Tim has touched a bit entries on Drinfel’d twist and the more general bialgebra cocycle a la Shahn Majid and I have added another kind of bialgebra cocycles, namely those defining the Gerstenhaber-Schack cohomology. I added a tag gebra to this post (cogebras, bigebras etc.).
Actually my main claim to fame on Drinfel’d twist was to link Shahn Majid’s name to his page. I was looking at this entry because of trying to sort out (i) what Turaev’s Hcobordism category ’really’ is (in HQFTs) when his ’target’ space is a $K(G,1)$. This corresponds to equivariant TQFTs. Various people have identified his algebraic models as being special types of Frobenius objects in a category of modules over the Drinfel’d double of $G$. (ii) trying to understand what might be the analogous construction when the target is more general, e.g. the classifying space of a crossed module. This should lead to a higher version of the double. (It relates back to ideas that I wanted to explore about 13 years ago, as the link that Freed and then Simon (Willerton) made between the double and the action groupoid of $G$ acting by conjugation on itself, have a distinctive simplicial perhaps quasicategory feeling. My plan back in 1998 was to test the Yetter construction of a TQFT from a crossed module pushing it to have ’coefficients’ in a simplicial group, and to handle extended TQFTs. (I copied the research proposal to my nLab page.))
The deformation of Turaev’s theory would then be potentially possible, though in what precise way I do not yet see.
Bressler had some unfinished and unpublished work in 2002 (before Willerton) in which he considered what is the general story behind the Drinfeld double and similar constructions and emphasised on a connection to cyclic homology and inertia groupoid (this motivated me to get a theorem on Drinfeld double which is a variant of what Hinich independently got and is now widely used as a motivation by Ben Zvi and others (unfortunately, I gave up writing it down and publishing once I heard from Bressler that Hinich got more or less the same theorem). But what is interesting in Bressler’s work is, among other claims, that not only SL(2, Z)-action but in fact SL(n,Z) action is possible to engineer in similar models. One just needs to iterate the inertia construction many times, and this on the other hand leads to higher stacks.
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